arX

iv:0

706.

1696

v1 [

hep-

ph]

12

Jun

2007

Minimal Walking Technicolor:

Set Up for Collider Physics

Roshan Foadi,∗ Mads T. Frandsen,† Thomas A. Ryttov,‡ and Francesco Sannino§

CERN Theory Division, CH-1211 Geneva 23, Switzerland and

University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark.

Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen, Denmark.

Abstract

Different theoretical and phenomenological aspects of the Minimal and Nonminimal Walking

Technicolor theories have recently been studied. The goal here is to make the models ready for

collider phenomenology. We do this by constructing the low energy effective theory containing

scalars, pseudoscalars, vector mesons and other fields predicted by the minimal walking theory.

We construct their self-interactions and interactions with standard model fields. Using the Wein-

berg sum rules, opportunely modified to take into account the walking behavior of the underlying

gauge theory, we find interesting relations for the spin-one spectrum. We derive the electroweak

parameters using the newly constructed effective theory and compare the results with the under-

lying gauge theory. Our analysis is sufficiently general such that the resulting model can be used

to represent a generic walking technicolor theory not at odds with precision data.

∗Electronic address: roshan@fysik.sdu.dk†Electronic address: toudal@nbi.dk‡Electronic address: ryttov@nbi.dk§Electronic address: sannino@fysik.sdu.dk

1

I. INTRODUCTION

Recently we have uncovered the phase diagram of strongly coupled theories [1, 2] in the

ladder approximation [3, 4], as a function of the number of flavors and colors, with Dirac

fermions transforming according to a given but arbitrary representation of the underlying

SU(N) gauge group. Further studies of the conformal window and its properties can be

found in [5, 6, 7].

We have also identified a number of strongly coupled theories which can dynamically

break the electroweak gauge symmetry and pass the precision tests. The systematic analysis

started in [8] and was extended in [1]. A related study can be found in [9]. First principle

numerical lattice computations have already initiated the study of the proposed conformal

window, and preliminary results just appeared [10] which seem to support the near conformal

(walking) behavior of the simplest theory. These are initial investigations on very small

lattices which encourage one to embark on a more serious study on larger lattices. The

simplest of these theories has fermions in the two index symmetric representation of the

gauge group, and is argued to already walk with only two Dirac flavors and two colors

[2]. Therefore, when used to break the electroweak symmetry dynamically, we will call it

the Minimal Walking Technicolor (MWT) model. This extension of the standard model

passes the electroweak precision constraints [8] while displaying some interesting features.

For example, it allows for a successful unification of the standard model couplings at the

one loop level [11]. Also interesting types of dark matter components [12, 13, 14] can be

envisioned. The walking dynamics was first introduced in [15, 16, 17, 18, 19] to explain the

breaking of the electroweak theory.

Here we take our phenomenological program one step closer to collider phenomenology

by constructing the theory at the underlying and effective Lagrangian level. We include the

relevant fields which can be discovered at collider experiments, with their self interactions

as well as the interactions with the standard model fields. We provide the link with the

underlying gauge theory via the time-honored Weinberg sum rules (WSR)s [20], in case of

running or walking dynamics. By running dynamics here we mean that the coupling constant

of the associated asymptotically free gauge theory has a dependence as function of energy

similar to the one in Quantum Chromo Dynamics. In the walking dynamics regime the

coupling constant is almost constant for a wide range of energy before resuming the running

2

behavior at very high energies, higher than the scale below which chiral symmetry breaking

occurs. In order to implement the walking dynamics we use and generalize the results

presented in [21]. This link allows us to provide important phenomenological relations for

the spectrum of the axial and vector type spin one mesons. We find that light vector mesons

(around 1 TeV) are compatible only within the walking regime if one requires simultaneously

that the underlying strongly coupled theory leads to an S parameter [22] smaller than the

one associated to the first technicolor models (which were based on an SU(3) gauge theory

with two Dirac techniflavors [23]). We also show how naturally the walking dynamics merges

into the running one.

Our analysis and effective Lagrangian are sufficiently general to be applicable to the

vast majority of models of dynamical electroweak symmetry breaking in agreement with the

precision tests, and for which a strongly coupled four dimensional gauge theory underlies

the dynamics. Since the global symmetry is SU(4) for the MWT, one can easily reduce our

model to the case of an SU(2)L×SU(2)R chiral symmetry. For certain values of the couplings

of the low energy effective theory - before imposing the generalized WSRs - one recovers the

BESS models [24].

Walking dynamics differs from the QCD-like running behavior because of a nearby in-

frared (IR) fixed point which dominates the low energy dynamics. The physics of the fixed

point theory per se is very interesting. If one assumes the existence of a theory with an actual

IR fixed point coupled to a non-conformal theory (such as the standard model), in the way

described recently by Georgi [25, 26], this leads to interesting phenomenology [27, 28, 29, 30].

The presence of a conformal symmetry signals itself in a way that formally resembles the

production of a non-integer number of massless invisible particles, where the non-integer

number is nothing but the scale dimension of the conformal-sector opertor, which is weakly

coupled to the standard model operators. We expect, however, following reference [31], that

the coupling with the standard model fields will push the unparticle sector away from the

IR fixed point. If this is the case, in practice one will observe a walking dynamics in certain

sectors of the theory, such as for example the electroweak symmetry breaking sector. Our

model should then be a reasonable description of a near conformal dynamics associated to

this sector.

The rather comprehensive model we are going to develop in the following sections has

been conceived in a way to ease its implementation on computer programs aiming to provide

3

interesting experimental signals for the physics at colliders.

II. THE UNDERLYING LAGRANGIAN FOR MINIMAL WALKING TECHNI-

COLOR

The new dynamical sector we consider, which underlies the Higgs mechanism, is an SU(2)

technicolor gauge theory with two adjoint technifermions [2]. The theory is asymptotically

free if the number of flavors Nf is less than 2.75.

The two adjoint fermions may be written as

QaL =

Ua

Da

L

, UaR , Da

R , a = 1, 2, 3 , (1)

with a being the adjoint color index of SU(2). The left handed fields are arranged in

three doublets of the SU(2)L weak interactions in the standard fashion. The condensate is

〈UU + DD〉 which correctly breaks the electroweak symmetry.

The model as described so far suffers from the Witten topological anomaly [32]. However,

this can easily be solved by adding a new weakly charged fermionic doublet which is a

technicolor singlet [8]. Schematically:

LL =

N

E

L

, NR , ER . (2)

In general, the gauge anomalies cancel using the following generic hypercharge assignment

Y (QL) =y

2, Y (UR, DR) =

(y + 1

2,y − 1

2

), (3)

Y (LL) =− 3y

2, Y (NR, ER) =

(−3y + 1

2,−3y − 1

2

), (4)

where the parameter y can take any real value [8]. In our notation the electric charge is

Q = T3 + Y , where T3 is the weak isospin generator. One recovers the SM hypercharge

assignment for y = 1/3.

To discuss the symmetry properties of the theory it is convenient to use the Weyl basis

for the fermions and arrange them in the following vector transforming according to the

4

fundamental representation of SU(4)

Q =

UL

DL

−iσ2U∗R

−iσ2D∗R

, (5)

where UL and DL are the left handed techniup and technidown, respectively and UR and

DR are the corresponding right handed particles. Assuming the standard breaking to the

maximal diagonal subgroup, the SU(4) symmetry spontaneously breaks to SO(4). Such a

breaking is driven by the following condensate

〈Qαi Q

βj ǫαβE

ij〉 = −2〈URUL +DRDL〉 , (6)

where the indices i, j = 1, . . . , 4 denote the components of the tetraplet of Q, and the Greek

indices indicate the ordinary spin. The matrix E is a 4 × 4 matrix defined in terms of the

2-dimensional unit matrix as

E =

0 1

1 0

. (7)

We follow the notation of Wess and Bagger [33] ǫαβ = −iσ2αβ and 〈Uα

LUR∗βǫαβ〉 =

−〈URUL〉. A similar expression holds for the D techniquark. The above condensate is

invariant under an SO(4) symmetry. This leaves us with nine broken generators with asso-

ciated Goldstone bosons.

Replacing the Higgs sector of the SM with the MWT the Lagrangian now reads:

LH → −1

4Fa

µνFaµν + iQLγµDµQL + iURγ

µDµUR + iDRγµDµDR

+iLLγµDµLL + iNRγ

µDµNR + iERγµDµER (8)

with the technicolor field strength Faµν = ∂µAa

ν − ∂νAaµ + gTCǫ

abcAbµAc

ν, a, b, c = 1, . . . , 3.

For the left handed techniquarks the covariant derivative is:

5

DµQaL =

(δac∂µ + gTCAb

µǫabc − i

g

2~Wµ · ~τδac − ig′

y

2Bµδ

ac)Qc

L . (9)

Aµ are the techni gauge bosons, Wµ are the gauge bosons associated to SU(2)L and Bµ is

the gauge boson associated to the hypercharge. τa are the Pauli matrices and ǫabc is the fully

antisymmetric symbol. In the case of right handed techniquarks the third term containing

the weak interactions disappears and the hypercharge y/2 has to be replaced according

to whether it is an up or down techniquark. For the left-handed leptons the second term

containing the technicolor interactions disappears and y/2 changes to −3y/2. Only the last

term is present for the right handed leptons with an appropriate hypercharge assignment.

III. LOW ENERGY THEORY FOR MWT

We construct the effective theory for MWT including composite scalars and vector bosons,

their self interactions, and their interactions with the electroweak gauge fields and the stan-

dard model fermions

A. Scalar Sector

The relevant effective theory for the Higgs sector at the electroweak scale consists, in

our model, of a composite Higgs and its pseudoscalar partner, as well as nine pseudoscalar

Goldstone bosons and their scalar partners. These can be assembled in the matrix

M =

[σ + iΘ

2+√2(iΠa + Πa)Xa

]E , (10)

which transforms under the full SU(4) group according to

M → uMuT , with u ∈ SU(4) . (11)

6

The Xa’s, a = 1, . . . , 9 are the generators of the SU(4) group which do not leave the vacuum

expectation value (VEV) of M invariant

〈M〉 = v

2E . (12)

Note that the notation used is such that σ is a scalar while the Πa’s are pseudoscalars. It

is convenient to separate the fifteen generators of SU(4) into the six that leave the vacuum

invariant, Sa, and the remaining nine that do not, Xa. Then the Sa generators of the SO(4)

subgroup satisfy the relation

SaE + E SaT = 0 , with a = 1, . . . , 6 , (13)

so that uEuT = E, for u ∈ SO(4). The explicit realization of the generators is shown in

appendix A.

Notice that it is necessary to introduce the “tilde” fields in the matrix M when realizing

the global symmetry linearly. In fact, it can easily be shown that the matrix

M =(σ2+ i

√2ΠaXa

)E

is not invariant in form under a general SU(4) transformation, but only under transforma-

tions of the unbroken SO(4) subgroup. This is in contrast to the case of an SU(2)L×SU(2)R

chiral group, whose minimal form involves a scalar Higgs and three pseudoscalar Goldstone

bosons only, but is similar to the case of an SU(3)L × SU(3)R chiral group. With the tilde

fields included, the matrix M is invariant in form under U(4)≡SU(4)×U(1)A, rather than

just SU(4). However the U(1)A axial symmetry is anomalous, and is therefore broken at the

quantum level.

The connection between the composite scalars and the underlying techniquarks can be

derived from the transformation properties under SU(4), by observing that the elements of

the matrix M transform like techniquark bilinears:

Mij ∼ Qαi Q

βj εαβ with i, j = 1 . . . 4. (14)

Using this expression, and the basis matrices given in appendix A, the scalar fields can

7

be related to the wavefunctions of the techniquark bound states. This gives the following

charge eigenstates:

v +H ≡ σ ∼ UU +DD , Θ ∼ i(Uγ5U +Dγ5D

),

A0 ≡ Π3 ∼ UU −DD , Π0 ≡ Π3 ∼ i(Uγ5U −Dγ5D

),

A+ ≡ Π1 − iΠ2

√2

∼ DU , Π+ ≡ Π1 − iΠ2

√2

∼ iDγ5U ,

A− ≡ Π1 + iΠ2

√2

∼ UD , Π− ≡ Π1 + iΠ2

√2

∼ iUγ5D ,

(15)

for the technimesons, and

ΠUU ≡ Π4 + iΠ5 +Π6 + iΠ7

2∼ UTCU ,

ΠDD ≡ Π4 + iΠ5 − Π6 − iΠ7

2∼ DTCD ,

ΠUD ≡ Π8 + iΠ9

√2

∼ UTCD ,

ΠUU ≡ Π4 + iΠ5 + Π6 + iΠ7

2∼ iUTCγ5U ,

ΠDD ≡ Π4 + iΠ5 − Π6 − iΠ7

2∼ iDTCγ5D ,

ΠUD ≡ Π8 + iΠ9

√2

∼ iUTCγ5D ,

(16)

for the technibaryons, where U ≡ (UL, DL)T and D ≡ (DL, DR)

T are Dirac technifermions,

and C is the charge conjugation matrix, needed to form Lorentz-invariant objects. To these

technibaryon charge eigenstates we must add the corresponding charge conjugate states (e.g.

ΠUU → ΠUU).

The electroweak subgroup can be embedded in SU(4), as explained in detail in [34].

Here SO(4) acts as a vectorial subgroup, in the sense that this is the diagonal subgroup

to which SU(4) is maximally broken. Based on this, we can say that the generators Sa,

with a = 1, 2, 3, form a vectorial SU(2) subgroup of SU(4), which is henceforth denoted by

SU(2)V, while S4 forms a U(1)V subgroup. The Sa generators, with a = 1, .., 4, together

with the Xa generators, with a = 1, 2, 3, generate an SU(2)L×SU(2)R×U(1)V algebra. This

8

is easily seen by changing genarator basis from (Sa, Xa) to (La, Ra), where

La ≡ Sa +Xa

√2

=

τa

20

0 0

, −RaT ≡ Sa −Xa

√2

=

0 0

0 − τaT

2

, (17)

with a = 1, 2, 3. The electroweak gauge group is then obtained by gauging SU(2)L, and the

U(1)Y subgroup of SU(2)R ×U(1)V, where

Y = −R3T +√2 YV S4 , (18)

and YV is the U(1)V charge. For example, from Eq. (3) and Eq. (4) we see that YV = y for

the techniquarks, and YV = −3y for the new leptons. As SU(4) spontaneously breaks to

SO(4), SU(2)L × SU(2)R breaks to SU(2)V. As a consequence, the electroweak symmetry

breaks to U(1)Q, where

Q =√2 S3 +

√2 YV S4 . (19)

Moreover the SU(2)V group, being entirely contained in the unbroken SO(4), acts as a

custodial isospin, which insures that the ρ parameter is equal to one at tree-level.

The electroweak covariant derivative for the M matrix is

DµM = ∂µM − i g[Gµ(y)M +MGT

µ (y)], (20)

where

g Gµ(YV) = g W aµ La + g′ Bµ Y

= g W aµ La + g′ Bµ

(−R3T +

√2 YV S4

). (21)

Notice that in the last equation Gµ(YV) is written for a general U(1)V charge YV, while in

Eq. (20) we have to take the U(1)V charge of the techniquarks, YV = y, since these are the

constituents of the matrix M , as explicitly shown in Eq. (14).

Three of the nine Goldstone bosons associated with the broken generators become the

longitudinal degrees of freedom of the massive weak gauge bosons, while the extra six Gold-

9

stone bosons will acquire a mass due to extended technicolor interactions (ETC) as well as

the electroweak interactions per se. Using a bottom up approach we will not commit to a

specific ETC theory but limit ourself to introduce the minimal low energy operators needed

to construct a phenomenologically viable theory. The new Higgs Lagrangian is

LHiggs =1

2Tr

[DµMDµM †

]− V(M) + LETC , (22)

where the potential reads

V(M) = −m2

2Tr[MM †] +

λ

4Tr

[MM †

]2+ λ′Tr

[MM †MM †

]

− 2λ′′[Det(M) + Det(M †)

], (23)

and LETC contains all terms which are generated by the ETC interactions, and not by the

chiral symmetry breaking sector. Notice that the determinant terms (which are renormal-

izable) explicitly break the U(1)A symmetry, and give mass to Θ, which would otherwise

be a massless Goldstone boson. While the potential has a (spontaneously broken) SU(4)

global symmetry, the largest global symmetry of the kinetic term is SU(2)L×U(1)R×U(1)V

(where U(1)R is the τ 3 part of SU(2)R), and becomes SU(4) in the g, g′ → 0 limit. Under

electroweak gauge transformations, M transforms like

M(x) → u(x; y) M(x) uT (x; y) , (24)

where

u(x; YV) = exp[iαa(x)La + iβ(x)

(−R3T +

√2 YV S4

)], (25)

and YV = y. We explicitly break the SU(4) symmetry in order to provide mass to the Gold-

stone bosons which are not eaten by the weak gauge bosons. We, however, preserve the full

SU(2)L×SU(2)R×U(1)V subgroup of SU(4), since breaking SU(2)R×U(1)V to U(1)Y would

result in a potentially dangerous violation of the custodial isospin symmetry. Assuming

parity invariance we write:

LETC =m2

ETC

4Tr

[MBM †B +MM †

]+ · · · , (26)

10

where the ellipses represent possible higher dimensional operators, and B ≡ 2√2S4 com-

mutes with the SU(2)L×SU(2)R×U(1)V generators.

The potential V(M) is SU(4) invariant. It produces a VEV which parameterizes the

techniquark condensate, and spontaneously breaks SU(4) to SO(4). In terms of the model

parameters the VEV is

v2 = 〈σ〉2 = m2

λ+ λ′ − λ′′, (27)

while the Higgs mass is

M2H = 2 m2 . (28)

The linear combination λ + λ′ − λ′′ corresponds to the Higgs self coupling in the SM. The

three pseudoscalar mesons Π±, Π0 correspond to the three massless Goldstone bosons which

are absorbed by the longitudinal degrees of freedom of the W± and Z boson. The remaining

six uneaten Goldstone bosons are technibaryons, and all acquire tree-level degenerate mass

through, not yet specified, ETC interactions:

M2ΠUU

=M2ΠUD

=M2ΠDD

= m2ETC . (29)

The remaining scalar and pseudoscalar masses are

M2Θ = 4v2λ′′

M2A± =M2

A0 = 2v2 (λ′ + λ′′) (30)

for the technimesons, and

M2eΠUU

=M2eΠUD

=M2eΠDD

= m2ETC + 2v2 (λ′ + λ′′) , (31)

for the technibaryons. To gain insight on some of the mass relations one can use [35].

11

B. Vector Bosons

The composite vector bosons of a theory with a global SU(4) symmetry are conveniently

described by the four-dimensional traceless Hermitian matrix

Aµ = Aaµ T a , (32)

where T a are the SU(4) generators: T a = Sa, for a = 1, . . . , 6, and T a+6 = Xa, for a =

1, . . . , 9. Under an arbitrary SU(4) transformation, Aµ transforms like

Aµ → u Aµ u† , where u ∈ SU(4) . (33)

Eq. (33), together with the tracelessness of the matrix Aµ, gives the connection with the

techniquark bilinears:

Aµ,ji ∼ Qα

i σµ

αβQβ,j − 1

4δjiQ

αkσ

µ

αβQβ,k . (34)

Then we find the following relations between the charge eigenstates and the wavefunctions

of the composite objects:

v0µ ≡ A3µ ∼ UγµU − DγµD , a0µ ≡ A9µ ∼ Uγµγ5U − Dγµγ5D

v+µ ≡ A1µ − iA2µ

√2

∼ DγµU , a+µ ≡ A7µ − iA8µ

√2

∼ Dγµγ5U

v−µ ≡ A1µ + iA2µ

√2

∼ UγµD , a−µ ≡ A7µ + iA8µ

√2

∼ Uγµγ5D

v4µ ≡ A4µ ∼ UγµU + DγµD ,

(35)

for the vector mesons, and

xµUU ≡ A10µ + iA11µ + A12µ + iA13µ

2∼ UTCγµγ5U ,

xµDD ≡ A10µ + iA11µ −A12µ − iA13µ

2∼ DTCγµγ5D ,

xµUD ≡ A14µ + iA15µ

√2

∼ DTCγµγ5U ,

sµUD ≡ A6µ − iA5µ

√2

∼ UTCγµD ,

(36)

for the vector baryons.

There are different approaches on how to introduce vector mesons at the effective La-

12

grangian level. At the tree level they are all equivalent. The main differences emerge when

exploring quantum corrections.

In the appendix we will show how to introduce the vector mesons in a way that renders

the following Lagrangian amenable to loop computations. Based on these premises, the

kinetic Lagrangian is:

Lkinetic = −1

2Tr

[WµνW

µν]− 1

4BµνB

µν − 1

2Tr

[FµνF

µν]+m2

A Tr[CµC

µ], (37)

where Wµν and Bµν are the ordinary field strength tensors for the electroweak gauge fields.

Strictly speaking the terms above are not only kinetic ones since the Lagrangian contains a

mass term as well as self interactions. The tilde on W a indicates that the associated states

are not yet the standard model weak triplets: in fact these states mix with the composite

vectors to form mass eigenstates corresponding to the ordinary W and Z bosons. Fµν is the

field strength tensor for the new SU(4) vector bosons,

Fµν = ∂µAν − ∂νAµ − ig [Aµ, Aν ] , (38)

and the vector field Cµ is defined by

Cµ ≡ Aµ − g

gGµ(y) . (39)

As shown in the appendix this is the appropriate linear combination to take which transforms

homogeneously under the electroweak symmetries:

Cµ(x) → u(x; y) Cµ(x) u(x; y)† , (40)

where u(x; YV) is given by Eq. (25). (Once again, the specific assignment YV = y, due to the

fact that the composite vectors are built out of techniquark bilinears.) The mass term in

Eq. (37) is gauge invariant (see the appendix), and gives a degenerate mass to all composite

vector bosons, while leaving the actual gauge bosons massless. (The latter acquire mass as

usual from the covariant derivative term of the scalar matrixM , after spontaneous symmetry

breaking.)

The Cµ fields couple with M via gauge invariant operators. Up to dimension four oper-

13

ators the Lagrangian is (see the appendix for a more general treatment):

LM−C = g2 r1 Tr[CµC

µMM †]+ g2 r2 Tr

[CµMCµTM †

]

+ i g r3 Tr[Cµ

(M(DµM)† − (DµM)M †

)]+ g2 s Tr [CµC

µ] Tr[MM †

]. (41)

The dimensionless parameters r1, r2, r3, s parameterize the strength of the interactions

between the composite scalars and vectors in units of g, and are therefore naturally expected

to be of order one. However, notice that for r1 = r2 = r3 = 0 the overall Lagrangian

possesses two independent SU(2)L×U(1)R×U(1)V global symmetries. One for the terms

involving M and one for the terms involving Cµ [63]. The Higgs potential only breaks the

symmetry associated with M , while leaving the symmetry in the vector sector unbroken.

This enhanced symmetry guarantees that all r-terms are still zero after loop corrections.

Moreover if one chooses r1, r2, r3 to be small the near enhanced symmetry will protect these

values against large corrections [24, 34].

We can also construct dimension four operators including only Cµ fields. These new

operators will not affect our analysis but will be relevant when investigating corrections to

the trilinear and quadrilinear gauge bosons interactions. We will include these terms in

appendix C.

C. Fermions and Yukawa Interactions

The fermionic content of the effective theory consists of the standard model quarks and

leptons, the new lepton doublet L = (N,E) introduced to cure the Witten anomaly, and a

composite techniquark-technigluon doublet.

We now consider the limit according to which the SU(4) symmetry is, at first, extended

to ordinary quarks and leptons. Of course, we will need to break this symmetry to accom-

modate the standard model phenomenology. We start by arranging the SU(2) doublets in

SU(4) multiplets as we did for the techniquarks in Eq. (5). We therefore introduce the four

14

component vectors qi and li,

qi =

uiL

diL

−iσ2uiR∗

−iσ2diR∗

, li =

νiL

eiL

−iσ2νiR∗

−iσ2eiR∗

, (42)

where i is the generation index. Note that such an extended SU(4) symmetry automatically

predicts the presence of a right handed neutrino for each generation. In addition to the

standard model fields there is an SU(4) multiplet for the new leptons,

L =

NL

EL

−iσ2NR∗

−iσ2ER∗

, (43)

and a multiplet for the techniquark-technigluon bound state,

Q =

UL

DL

−iσ2U∗R

−iσ2D∗R

. (44)

With this arrangement, the electroweak covariant derivative for the fermion fields can be

written

Dµ = ∂µ − i g Gµ(YV) , (45)

where YV = 1/3 for the quarks, YV = −1 for the leptons, YV = −3y for the new lepton

doublet, and YV = y for the techniquark-technigluon bound state. One can check that these

charge assignments give the correct electroweak quantum numbers for the standard model

fermions. In addition to the covariant derivative terms, we should add a term coupling Q

to the vector field Cµ, which transforms globally under electroweak gauge transformations.

Such a term naturally couples the composite fermions to the composite vector bosons which

15

otherwise would only feel the week interactions. Based on this, we write the following gauge

part of the fermion Lagrangian:

Lfermion = i qiασµ,αβDµq

iβ + i l

i

ασµ,αβDµl

iβ + i Lασ

µ,αβDµLβ + i Qασµ,αβDµQβ

+ x Qασµ,αβCµQβ (46)

The terms coupling the standard model fermions or the new leptons to Cµ are in general

not allowed. In fact under electroweak gauge transformations any four-component fermion

doublet ψ transforms like

ψ → u(x; YV) ψ , (47)

and from Eq. (40) we see that a term like ψασµ

αβCµψ

βis only invariant if YV = y. Then we can

distinguish two cases. First, we can have y 6= 1/3 and y 6= −1, in which case ψασµ

αβCµψ

βis

only invariant for ψ = Q. Interaction terms of the standard model fermions with components

of Cµ are still possible, but these would break the SU(4) chiral simmetry even in the limit

in which the electroweak gauge interactions are switched off. Second, we can have y = 1/3

or y = −1. Then ψασµ

αβCµψ

βis not only invariant for ψ = Q, but also for either ψ = qi

or ψ = li, respectively. In the last two cases, however, the corresponding interactions are

highly suppressed, since these give rise to anomalous couplings of the light fermions with

the standard model gauge bosons, which are tightly constrained by experiments.

We now turn to the fundamental issue of providing masses to ordinary fermions. Many

extensions of technicolor have been suggested in the literature to address this problem. Some

of the extensions use another strongly coupled gauge dynamics, others introduce fundamen-

tal scalars. Many variants of the schemes presented above exist and a review of the major

models is the one by Hill and Simmons [36]. At the moment there is not yet a consensus

on which is the correct ETC. To keep the number of fields minimal we make the most eco-

nomical ansatz, i.e. we parameterize our ignorance about a complete ETC theory by simply

coupling the fermions to our low energy effective Higgs. This simple construction minimizes

the flavor changing neutral currents problem. It is worth mentioning that it is possible to

engineer a schematic ETC model proposed first by Randall in [37] and adapted for the MWT

in [38] for which the effective theory presented in the main text can be considered a minimal

16

description. [64]

Depending on the value of y for the techniquarks, we can write different Yukawa inter-

actions which couple the standard model fermions to the matrix M . Let ψ denote either qi

or li. If ψ and the techniquark multiplets Qa have the same U(1)V charge, then the Yukawa

term

− ψTM∗ψ + h.c. , (48)

is gauge invariant, as one can check explicitly from Eq. (24) and Eq. (47). Otherwise, if ψ

and Qa have different U(1)V charges, we can only write a gauge invariant Lagrangian with

the off-diagonal terms of M , which contain the Higgs and the Goldstone bosons:

− ψTM∗off ψ + h.c. . (49)

In fact Moff has no U(1)V charge, since

S4Moff +MoffS4T = 0 , (50)

The last equation implies that the U(1)V charges of ψT and ψ cancel in Eq. (49). The latter

is actually the only viable Yukawa Lagrangian for the new leptons, since the corresponding

U(1)V charge is YV = −3y 6= y, and for the ordinary quarks, since Eq. (48) contains qq terms

which are not color singlets.

We notice however that neither Eq. (48) nor Eq. (49) are phenomenologically viable yet,

since they leave the SU(2)R subgroup of SU(4) unbroken, and the corresponding Yukawa

interactions do not distinguish between the up-type and the down-type fermions. In order

to prevent this feature, and recover agreement with the experimental input, we break the

SU(2)R symmetry to U(1)R by using the projection operators PU and PD, where

PU =

1 0

0 1+τ3

2

, PD =

1 0

0 1−τ3

2

. (51)

17

Then, for example, Eq. (48) should be replaced by

− ψT (PUM∗PU)ψ − ψT (PDM

∗PD)ψ + h.c. . (52)

For illustration we distinguish two different cases for our analysis, y 6= −1 and y = −1,

and write the corresponding Yukawa interactions:

(i) y 6= −1. In this case we can only form gauge invariant terms with the standard model

fermions by using the off-diagonal M matrix. Allowing for both N − E and U − D mass

splitting, we write

LYukawa = − yiju qiT (PUM∗offPU) q

j − yijd qiT (PDM∗offPD) q

j

− yijν liT (PUM∗offPU) l

j − yije liT (PDM∗offPD) l

j

− yN LT (PUM∗offPU)L− yE LT (PDM

∗offPD)L

− yeUQT (PUM

∗PU) Q− y eDQT (PDM

∗PD) Q + h.c. , (53)

where yiju , yijd , y

ijν , y

ije are arbitrary complex matrices, and yN , yE , yeU

, y eDare complex

numbers.

Note that the underlying strong dynamics already provides a dynamically generated mass

term for Q of the type:

k QTM∗Q+ h.c. , (54)

with k a dimensionless coefficient of order one and entirely fixed within the underlying

theory. The splitting between the up and down type techniquarks is due to physics beyond

the technicolor interactions [65]. Hence the Yukawa interactions for Q must be interpreted

as already containing the dynamical generated mass term.

(ii) y = −1. In this case we can form gauge invariant terms with the standard model

leptons and the full M matrix:

LYukawa = − yiju qiT (PUM∗offPU) q

j − yijd qiT (PDM∗offPD) q

j

− yijν liT (PUM∗PU) l

j − yije liT (PDM∗PD) l

j

− yN LT (PUM∗offPU)L− yE LT (PDM

∗offPD)L

− yeUQT (PUM

∗PU) Q− y eDQT (PDM

∗PD) Q + h.c. . (55)

18

Here we are assuming Dirac masses for the neutrinos, but we can easily add also Majorana

mass terms. At this point one can exploit the symmetries of the kinetic terms to induce a

GIM mechanism, which works out exactly like in the standard model. Therefore, in both

Eq. (53) and Eq. (55) we can assume yiju , yijd , y

ijν , y

ije to be diagonal matrices, and replace

the diL and νiL fields, in the kinetic terms, with V ijq d

jL and V ij

l νjL, respectively, where Vq and

Vl are the mixing matrices.

When y = −1 Q has the same quantum numbers of the ordinary leptons, except for

the technibaryon number. If the technibaryon number is violated they can mix with the

ordinary leptons behaving effectively as a fourth generation leptons (see Eq. (55)). However

this will reintroduce, in general, anomalous couplings with intermediate gauge bosons for

the ordinary fermions and hence we assume the mixing to be small.

IV. WEINBERG SUM RULES AND ELECTROWEAK PARAMETERS

The effective theory described until now has a number of free parameters which are fixed

once the associated underlying dynamics is specified. Conversely, the low effective theory

introduced above encompasses different underlying realizations with the same symmetry

pattern. In the following we partially reduce the arbitrariness of the Lagrangian by assuming

that the underlying dynamcics is the one of a four dimensional asymptotically free gauge

theory, with only fermionic matter fields transforming according to a given but otherwise

arbitrary representation of the gauge group. The MWT is automatically part of this set of

theories. We will use theWeinberg sum rules (WSR) as main ingredient to reduce the number

of unknown parameters in the theory. We will then compute the universal corrections in

the effective theory and compare these to the ones estimated in the underlying theory. The

latter step is taken only for the MWT theory but it straightforwardly generalizes to any

walking theory. We also use the results found in [21] which allow us to treat walking and

running theories in a unified way.

Weinberg sum rules

Our effective theory is meant to be associated to an underlying strongly coupled theory.

Hence we relate some of the parameters in the effective theory via the WSR. These are linked

19

to the two point vector-vector minus axial-axial vacuum polarization amplitude, which is

known to be sensitive to chiral symmetry breaking. We define

iΠa,bµν (q) ≡

∫d4x e−iqx

[< Ja

µ,V (x)Jbν,V (0) > − < Ja

µ,A(x)Jbν,A(0) >

], (56)

within the underlying strongly coupled gauge theory, where

Πa,bµν (q) =

(qµqν − gµνq

2)δabΠ(q2) . (57)

Here a, b = 1, ..., N2f −1, label the flavor currents and the SU(Nf) generators are normalized

according to Tr[TaTb

]= (1/2)δab. The function Π(q2) obeys the unsubtracted dispersion

relation1

π

∫ ∞

0

dsImΠ(s)

s +Q2= Π(Q2) , (58)

where Q2 = −q2 > 0, and the constraint −Q2Π(Q2) > 0 holds for 0 < Q2 < ∞ [45].

The discussion above is for the standard chiral symmetry breaking pattern SU(Nf )×SU(Nf )→SU(Nf) but it is generalizable to any breaking pattern.

Since we are imagining the underlying theory to be asymptotically free, the behavior of

Π(Q2) at asymptotically high momenta is the same as in ordinary QCD, i.e. it scales like

Q−6 [46]. Expanding the left hand side of the dispersion relation thus leads to the two

conventional spectral function sum rules

1

π

∫ ∞

0

ds ImΠ(s) = 0 and1

π

∫ ∞

0

ds s ImΠ(s) = 0 . (59)

Walking dynamics affects only the second sum rule [21] which is more sensitive to large

but not asymptotically large momenta due to fact that the associated integrand contains an

extra power of s.

We now saturate the absorptive part of the vacuum polarization. We follow reference [21]

and hence divide the energy range of integration in three parts. The resonance part which

we approximate by the vector and axial mesons as well as the Goldstone bosons. The second

one (the continuum region) where the walking dynamics takes over and which extends up

to the scale above which the underlying coupling constant drops like in a QCD-like theory.

20

The first WSR implies:

F 2V − F 2

A = F 2π , (60)

where F 2V and F 2

A are the vector and axial mesons decay constants. This sum rule holds for

walking and running dynamics. A more general representation of the resonance spectrum

would replace the left hand side of this relation with a sum over vector and axial states.

The second sum rule receives important contributions from throughout the near conformal

region and can be expressed in the form of:

F 2VM

2V − F 2

AM2A = a

8π2

d(R)F 4π , (61)

where a is expected to be positive and O(1) and d(R) is the dimension of the representation

of the underlying fermions. As in the case of the first sum rule, a more general resonance

spectrum will lead to a left-hand side with a sum over vector and axial states. In either case,

the conformal region enhances the vector piece relative to the axial. We have generalized the

result of reference [21] to the case in which the fermions belong to a generic representation

of the gauge group. In the case of running dynamics the right-hand side of the previous

equation vanishes.

Relating WSRs to the Effective Theory & S parameter

The S parameter is related to the absorptive part of the vector-vector minus axial-axial

vacuum polarization as follows:

S = 4

∫ ∞

0

ds

sImΠ(s) = 4π

[F 2V

M2V

− F 2A

M2A

], (62)

where ImΠ is obtained from ImΠ by subtracting the Goldstone boson contribution.

Other attempts to estimate the S parameter for walking technicolor theories have been

made in the past [47] showing reduction of the S parameter. S has also been evaluated using

computations inspired by the original AdS/CFT correspondence [48] in [49, 50, 51, 52, 53].

Very recently Kurachi, Shrock and Yamawaki [54] have further confirmed the results

presented in [21] with their computations tailored for describing four dimensional gauge

theories near the conformal window. The present approach [21] is more physical since it is

21

based on the nature of the spectrum of states associated directly to the underlying gauge

theory.

Note that in this work we are taking a rather conservative approach in which the S

parameter, although reduced with respect to the case of a running theory, is positive and

not very small. After all, other sectors of the theory such as new leptons can further reduce

or even offset a positive value of S due solely to the technicolor theory.

In our effective theory the S parameter is directly proportional to the parameter r3 via:

S =8π

g2χ (2− χ) , with χ =

v2g2

2M2A

r3 , (63)

where we have expanded in g/g and kept only the leading order. The full expression can be

found in appendix D. We can now use the sum rules to relate r3 to other parameters in the

theory for the running and the walking case. Within the effective theory we deduce:

F 2V =

(1− χ

r2r3

)2M2

A

g2=

2M2V

g2, F 2

A = 2M2

A

g2(1− χ)2 , F 2

π = v2(1− χ r3) . (64)

Hence the first WSR reads:

1 + r2 − 2r3 = 0 , (65)

while the second:

(r2 − r3)(v2g2(r2 + r3)− 4M2

A) = a16π2

d(R)v2 (1− χ r3)

2 . (66)

To gain analytical insight we consider the limit in which g is small while g/g is still much

smaller than one. To leading order in g the second sum rule simplifies to:

r3 − r2 = a4π2

d(R)

v2

M2A

, (67)

Together with the first sum rule we find:

r2 = 1− 2t , r3 = 1− t , (68)

22

with

t = a4π2

d(R)

v2

M2A

. (69)

The approximate S parameter reads.

S = 8πv2

M2A

(1− t) . (70)

A small value of a provides a large and positive t rendering S smaller than expected in a

running theory. In the next subsection we will make a similar analysis without taking the

limit of small g.

Axial-Vector Spectrum via WSRs

It is is interesting to determine the relative vector to axial spectrum as function of one of

the two masses, say the axial one, for a fixed value of the S parameter meant to be associated

to a given underlying gauge theory.

For a running type dynamics (i.e. a = 0) the two WSRs force the vector mesons to be

quite heavy (above 3 TeV) in order to have a relatively low S parameter (S ≃ 0.1). This

can be seen directly from Eq. (63) in the running regime, where r2 = r3 = 1. This leads to

M2A &

8πv2

S, (71)

which corresponds to MA & 3.6 TeV, for S ≃ 0.11. Perhaps a more physical way to express

this is to say that it is hard to have an intrinsically small S parameter for running type

theories. By small we mean smaller than the scaled up technicolor version of QCD with

two techniflavors, in which S ≃ 0.3. In Figure 1 we plot the difference between the axial

and vector mass as function of the axial mass, for S ≃0.11. Since Eq. (71) provides a lower

bound for MA, this plot shows that in the running regime the axial mass is always heavier

than the vector mass. In fact the M2A −M2

V difference is proportional to r2, with a positive

proportionality factor (see the appendix), and r2 = 1 in the running regime.

When considering the second WSR modified by the walking dynamics, we observe that it

is possible to have quite light spin one vector mesons compatible with a small S parameter.

23

3600 3800 4000 4200 4400 4600 4800 5000

MA HGeVL

0

500

1000

1500

2000

MA

-M

VHG

eVL

103

S`

= 1

MA t 3597

FIG. 1: In the picture above we have set 103S = 1, corresponding to S ≃ 0.11. In the appendix

we have provided the relation between S and the traditional S. Here we have imposed the first

and the second WSR for a = 0. This corresponds to an underlying gauge theory with a standard

running behavior of the coupling constant.

We numerically solve the first and second WSR in presence of the contribution due to walking

in the second sum rule. The results are summarized in Figure 2. As for the running case

we set again S ≃ 0.11. This value is close to the estimate in the underlying MWT [66].

The different curves are obtained by varying g from one (the thinnest curve) to eight (the

thickest curve). We plot the allowed values of MA−MV as function of MA in the left panel,

having imposed only the first sum rule. In the right panel we compute the corresponding

value that a should assume as function of MA in order for the second WSR to be satisfied

in the walking regime as well. For any given underlying gauge theory all of the values of the

parameters are fixed and our computation shows that it is possible to have walking theories

with light vector mesons and a small S parameter. Such a scenario needs a positive value

of a, together with an axial meson lighter than its associated vector meson for a between

zero and four, when the axial vector mass is lighter than approximatively 2.5 TeV. However

for spin one fields heavier than roughly 2.5 TeV and with still a positive a one has an axial

meson heavier than the vector one. A degenerate spectrum allows for a small S but with

relatively large values of a and spin one masses around 2.5 TeV. We observe that a becomes

zero (and eventually negative) when the vector spectrum becomes sufficiently heavy. In

other words we recover the running behavior for large masses of spin-one fields. Although

in the plot we show negative values of a one should stop the analysis after having reached

a zero value of a. In fact, for masses heavier than roughly 3.5 TeV the second WSR for the

24

0 1000 2000 3000 4000

MA HGeVL

-1250

-1000

-750

-500

-250

0

250

MA

-M

VHG

eVL

0 1000 2000 3000 4000

MA HGeVL

-4

-2

0

2

4

a

FIG. 2: In the two pictures above we have set 103S = 1, corresponding to S ≃ 0.11, and the

different curves are obtained by varying g from one (the thinnest curve) to eight (the thickest

curve). We have imposed the first WSR. Left Panel: We plot the allowed values of MA −MV as

function of MA. Right Panel: We compute the value that a should assume as function of MA in

order for the second WSR to be satisfied in the walking regime. Note that a is expected to be

positive or zero.

running behavior, i.e. a = 0, is enforced.

Our results are general and help elucidating how different underlying dynamics will man-

ifest itself at LHC. Any four dimensional strongly interacting theory replacing the Higgs

mechanism, with two Dirac techniflavors gauged under the electroweak theory, is expected

to have a spectrum of the low lying vector resonances like the one presented above.

We provide the relation between S and the recently proposed modification of the elec-

troweak parameters [55] in appendix D. In the same appendix we explicitly compute the

remaining parameters within our effective theory and check that they do not lead to further

constraints given the current status of the precision measurements.

Vanishing S via New Leptons

Although we have already studied the effects of the lepton family on the electroweak

parameters in [8], we summarize here the main results in Figure 3. The ellipses represent

the 68% confidence region for the S and T parameters. The upper ellipse is for a reference

Higgs mass of the order of 1 TeV while the lower curve is for a light Higgs with mass around

114 GeV. The contribution from the MWT theory per se and of the leptons as function of

the new lepton masses is expressed by the dark grey region. The left panel has been obtained

using a SM type hypercharge assignment while the right hand graph is for y = 1. In both

25

-0.2 -0.1 0 0.1 0.2

S

0

0.1

0.2

0.3

0.4

0.5

T

-0.2 -0.1 0 0.1 0.2

S

0

0.1

0.2

0.3

0.4

0.5

T

FIG. 3: The ellipses represent the 68% confidence region for the S and T parameters. The upper

ellipse is for a reference Higgs mass of the order of a TeV, the lower curve is for a light Higgs

with mass around 114 GeV. The contribution from the MWT theory per se and of the leptons as

function of the new lepton masses is expressed by the dark grey region. The left panel has been

obtained using a SM type hypercharge assignment while the right hand graph is for y = 1.

pictures the regions of overlap between the theory and the precision contours are achieved

when the upper component of the weak isospin doublet is lighter than the lower component.

The opposite case leads to a total S which is larger than the one predicted within the new

strongly coupled dynamics per se. This is due to the sign of the hypercharge for the new

leptons. The mass range used in the plots, in the case of the SM hypercharge assignment is

100 − 1000 GeV for the new electron and 50 − 800 GeV for the new Dirac neutrino, while

it is 100 − 800 and 100 − 1000 GeV respectively for the y = 1 case. The plots have been

obtained assuming a Dirac mass for the new neutral lepton (in the case of a SM hypercharge

assignment). The analysis for the Majorana mass case has been performed in [12] where one

can again show that it is easy to be within the 68% contours.

The contour plots we have drawn take into account the new values of the top mass which

has dropped dramatically since we last compared our theory in [8] to the experimental data

[61].

26

V. CONCLUSIONS

We have provided a comprehensive extension of the standard model which embodies

(minimal) walking technicolor theories and their interplay with the standard model parti-

cles. Our extension of the standard model features all of the relevant low energy effective

degrees of freedom linked to our underlying minimal walking theory. These include scalars,

pseudoscalars as well as spin one fields. The bulk of the Lagrangian has been spelled out.

The link with underlying strongly coupled gauge theories has been achieved via the time-

honored Weinberg sum rules. The modification of the latter according to walking has been

taken into account. We have also analyzed the case in which the underlying theory behaves

like QCD rather than being near an infrared fixed point. This has allowed us to gain insight

on the spectrum of the spin one fields which is an issue of phenomenological interest. In the

appendix we have: i) provided the explicit construction of all of the SU(4) generators, ii)

shown how to construct the effective Lagrangian in a way which is amenable to quantum

corrections, iii) shown the explicit form of the mass matrices for all of the particles, iv) pro-

vided a summary of all of the relevant electroweak parameters and their explicit dependence

on the coefficients of our effective theory.

We have introduced the model in a format which is, hopefully, user friendly for collider

phenomenology.

Acknowledgments

We have benefitted from discussions with A. Belyaev, S. Catterall, D.D. Dietrich, C.

Kouvaris, F. Krauss, J. Schechter and K. Tuominen. The work of R.F., M.T.F. and F.S. is

supported by the Marie Curie Excellence Grant under contract MEXT-CT-2004-013510.

APPENDIX A: GENERATORS

It is convenient to use the following representation of SU(4)

Sa =

A B

B† −AT

, X i =

C D

D† CT

, (A1)

27

where A is hermitian, C is hermitian and traceless, B = −BT and D = DT . The S are also a

representation of the SO(4) generators, and thus leave the vacuum invariant SaE+EST = 0 .

Explicitly, the generators read

Sa =1

2√2

τ

a 0

0 −τaT

, a = 1, . . . , 4 , (A2)

where a = 1, 2, 3 are the Pauli matrices and τ 4 = 1. These are the generators of SUV (2)×UV (1).

Sa =1

2√2

0 Ba

Ba† 0

, a = 5, 6 , (A3)

with

B5 = τ 2 , B6 = iτ 2 . (A4)

The rest of the generators which do not leave the vacuum invariant are

X i =1

2√2

τ

i 0

0 τ iT

, i = 1, 2, 3 , (A5)

and

X i =1

2√2

0 Di

Di† 0

, i = 4, . . . , 9 , (A6)

with

D4 = 1 , D6 = τ 3 , D8 = τ 1 ,

D5 = i1 , D7 = iτ 3 , D9 = iτ 1 .(A7)

The generators are normalized as follows

Tr[SaSb

]=

1

2δab , ,Tr

[X iXj

]=

1

2δij , Tr

[X iSa

]= 0 . (A8)

28

TABLE I: Field content

G G′

M R 1

N

Aµ 1 Adj

APPENDIX B: VECTOR MESONS AS GAUGE FIELDS

We show how to rewrite the vector meson Lagrangian in a gauge invariant way. We assume

the scalar sector to transform according to a given but otherwise arbitrary representation of

the flavor symmetry group G. This is a straightforward generalization of the Hidden Local

Gauge symmetry idea [57, 58], used in a similar context for the BESS models [24]. At the

tree approximation this approach is identical to the one introduced first in [59, 60].

1. Introducing Vector Mesons

Let us start with a generic flavor symmetry group G under which a scalar field M trans-

forms globally in a given, but generic, irreducible representation R. We also introduce an

algebra valued one-form A = Aµdxµ taking values in a copy of the algebra of the group G,

call it G′, i.e.

Aµ = AaµT

a , with T a ∈ A(G′) . (B1)

At this point the full group structure is the semisimple group G×G′. M does not transform

under G′. Given that M and A belong to two different groups we need another field to

connect the two. We henceforth introduce a new scalar field N transforming according to

the fundamental of G and to the antifundamental of G′. We then upgrade A to a gauge field

over G′. The covariant derivative for N is:

DµN = ∂µN + i g N Aµ . (B2)

29

We now force N to acquire the following vacuum expectation value

〈N ij〉 = δij v

′ , (B3)

which leaves the diagonal subgroup - denoted with GV - of G × G′ invariant. Clearly GV

is a copy of G. Note that it is always possible to arrange a suitable potential term for N

leading to the previous pattern of symmetry breaking. v/v′ is expected to be much less than

one and the unphysical massive degrees of freedom associated to the fluctuations of N will

have to be integrated out. The would-be Goldstone bosons associated to N will become the

longitudinal components of the massive vector mesons.

To connect A to M we define the one-form transforming only under G via N which - in

the deeply spontaneously broken phase of N - reads:

Tr[NN †]

dim(F )Pµ =

DµNN† −NDµN

†

2 ig, Pµ → uPµu

† , (B4)

with u being an element of G and dim(F ) the dimension of the fundamental representation

of G. When evaluating Pµ on the vacuum expectation value for N we recover Aµ:

〈Pµ〉 = Aµ . (B5)

At this point it is straightforward to write the Lagrangian containing N , M and A and their

self-interactions. Being in the deeply broken phase of G×G′ down to GV we count N as a

dimension zero field. This is consistent with the normalization for Pµ.

The kinetic term of the Lagrangian is:

Lkinetic = −1

2Tr [FµνF

µν ] +1

2Tr

[DNDN †

]+

1

2Tr

[∂M∂M †

]. (B6)

The second kinetic term will provide a mass to the vector mesons. Besides the potential

terms for M and N there is another part of the Lagrangian which is of interest to us. This

is the one mixing P and M . Up to dimension four and containing at most two powers of P

30

and M this is:

LP−M = g2 r1 Tr[PµP

µMM †]+ g2 r2 Tr

[PµMP µTM †

]

+ i g r3 Tr[Pµ

(M(DµM)† − (DµM)M †

)]+ g2 s Tr [PµP

µ] Tr[MM †

]. (B7)

The dimensionless parameters r1, r2, r3, s parameterize the strength of the interactions

between the composite scalars and vectors in units of g, and are therefore expected to be

of order one. We have assumed M to belong to the two index symmetric representation

of a generic G= SU(N). It is straightforward to generalize the previous terms to the case

of an arbitrary representation R with respect to any group G. Further higher derivative

interactions including N can be included systematically.

2. Further Gauging of G

In this case we add another gauge field Gµ taking values in the algebra of G. We then

define the correct covariant derivatives for M and N . For N , for example, we have:

DµN = ∂µN − i g GµN + i g N Aµ . (B8)

Evaluating the previous expression on the vacuum expectation value of N we recover the

field Cµ introduced in the text. To be more precise we need to use Pµ again but with the

covariant derivative for N replaced by the one in the equation above.

TABLE II: Field content

G G′

M R 1

N

Aµ 1 Adj

Gµ Adj 1

31

APPENDIX C: EFFECTIVE LAGRANGIAN AND MASS MATRICES

In this section we summarize and generalize the effective Lagrangians for the scalar and

vector sectors, and include the explicit mass matrices for the mixings of the composite

vectors with the fundamental gauge fields.

a. Scalar Sector

The composite scalars are assembled in the matrix M of Eq. (10). In terms of the mass

eigenstates this reads

M =

iΠUU + ΠUUiΠUD + ΠUD√

2

σ + iΘ+ iΠ0 +A0

2

iΠ+ +A+

√2

iΠUD + ΠUD√2

iΠDD + ΠDDiΠ− +A−

√2

σ + iΘ− iΠ0 −A0

√2

σ + iΘ+ iΠ0 +A0

2

iΠ− +A−

√2

iΠUU + ΠUU

iΠUD + ΠUD√2

iΠ+ +A+

√2

σ + iΘ− iΠ0 −A0

2

iΠUD + ΠUD√2

iΠDD + ΠDD

,

(C1)

where σ = v+H . The Lagrangian for the Higgs sector, including the spontaneously broken

potential, and the ETC mass term for the uneaten Goldstone bosons, is

LHiggs =1

2Tr

[DµMDµM †

]+m2

2Tr[MM †]

− λ

4Tr

[MM †

]2 − λ′Tr[MM †MM †

]+ 2λ′′

[Det(M) + Det(M †)

]

+m2

ETC

4Tr

[MBM †B +MM †

], (C2)

where the covariant derivative is given by Eq. (20).

32

b. Vector Sector

In terms of the charge eigenstates the matrix Aµ is

Aµ =

a0µ + v0µ + v4µ

2√2

a+µ + v+µ

2

xµUU√2

xµUD + sµUD

2

a−µ + v−µ

2

−a0µ − v0µ + v4µ

2√2

xµUD − sµUD

2

xµDD√2

xµUU√2

xµUD

− sµUD

2

a0µ − v0µ − v4µ

2√2

a−µ − v−µ

2

xµUD

+ sµUD

2

xµDD√2

a+µ − v+µ

2

−a0µ + v0µ − v4µ

2√2

. (C3)

The most general Lagrangian for the gauge and vector fields can be conveniently written

using the N and Pµ fields of appendix B. Demanding CP invariance, and including terms

up to dimension four, we have

Lvector = − 1

2Tr

[WµνW

µν]− 1

4Tr

[BµνB

µν]− 1

2Tr

[FµνF

µν]

+1

2Tr

[DµN (DµN)†

]+

1

2Tr

[DµM (DµM)†

]

+ g2 a1 Tr[PµP

µ]2

+ g2 a2 Tr[PµP

µPνPν]+ g2 a3 Tr

[PµPνP

µP ν]

− i g b Tr[[Pµ, Pν ]NF

µνN †]

+ g2 r1 Tr[PµP

µMM †]+ g2 r2 Tr

[PµMP µTM †

]

+ i g r3 Tr[Pµ

(M(DµM)† − (DµM)M †

) ]+ g2 s Tr

[PµP

µ]Tr

[MM †

],

(C4)

where the field strength tensor F µν is given by Eq. (38), and the covariant derivatives of

M and N are respectively given by Eq. (20) and Eq. (B8). Notice that we have excluded

the terms iTr [[Pµ, Pν ]Gµν ] and Tr

[NFµνN

†Gµν]with order one couplings. There terms in

the limit of no weak interactions are reserved solely to technicolor interactions. Here Gµν

contains W µν and Bµν .

The covariant derivative terms give rise to mass terms for the charged and neutral vector

33

bosons:

Lmass =(W−

µ v−µ a−µ

)M2

C

W+µ

v+µ

a+µ

+

1

2

(Bµ W 3

µ v0µ a0µ v4µ

)M2

N

Bµ

W 3µ

v0µ

a0µ

v4µ

, (C5)

where

M2

C =

g2M2V (1 + ω)

g2−gM

2V√2g

−gM2A(1− χ)√

2g

−gM2V√

2gM2

V 0

−gM2A(1− χ)√

2g0 M2

A

, (C6)

M2

N =

g′2M2V (1 + 2y2 + ω)

g2−gg

′M2V ω

g2−g

′M2V√

2g

g′M2A(1− χ)√2g

−g′M2

V (2y)√2g

−gg′M2

V ω

g2g2M2

V (1 + ω)

g2−gM

2V√

2g−gM

2A(1− χ)√

2g0

−g′M2

V√2g

−gM2V√2g

M2V 0 0

g′M2A(1− χ)√2g

−gM2A(1− χ)√

2g0 M2

A 0

−g′M2

V (2y)√2g

0 0 0 M2V

.

(C7)

Here MV and MA are the masses of the vector and axial vector bosons in absence of elec-

34

troweak interactions, and are related by

M2A =M2

V +1

2v2g2r2 . (C8)

The parameters ω and χ are defined by

ω ≡ v2g2

4M2V

(1 + r2 − 2r3)

χ ≡ v2g2

2M2A

r3 , (C9)

where χ has been used already in Eq. (63).

The vector baryons do not mix with the fundamental gauge fields and thus their masses

do not receive tree-level electroweak corrections. Therefore, xUU , xUD, and xDD are axial

mass eigenstates, and sUD is a vector mass eigenstate:

MxUU=MxUD

=MxDD=MA ,

MsUD=MV . (C10)

APPENDIX D: UNIVERSAL ELECTROWEAK CORRECTIONS

Any extension of the Standard Model cannot be at odds with the precision electroweak

data. Universal corrections to the Standard Model, i.e. corrections to the self-energies of

the vector bosons, may be encoded in a total of 7 parameters following reference [55]. We

show the relation with the ones presented in the main text [22] and use our newly built

effective theory to explicitly evaluate the corrections within the MWT. Let Q2 ≡ −q2 be

the Euclidean transferred momentum, and denote derivatives with respect to −Q2 with a

35

prime. Then we have the following definitions [55]:

S ≡ g2 Π′W 3B(0) , (D1)

T ≡ g2

M2W

[ΠW 3W 3(0)−ΠW+W−(0)] , (D2)

W ≡ g2M2W

2[Π′′

W 3W 3(0)] , (D3)

Y ≡ g′2M2W

2[Π′′

BB(0)] , (D4)

U ≡ −g2 [Π′W 3W 3(0)− Π′

W+W−(0)] , (D5)

V ≡ g2M2W

2[Π′′

W 3W 3(0)− Π′′W+W−(0)] , (D6)

X ≡ gg′M2W

2Π′′

W 3B(0) . (D7)

Here ΠV (Q2) with V = {W 3B, W 3W 3, W+W−, BB} represents the self-energy of the vec-

tor bosons. Here the electroweak couplings are the ones associated to the physical elec-

troweak gauge bosons:

1

g2≡ Π′

W+W−(0) ,1

g′2≡ Π′

BB(0) , (D8)

while GF is

1√2GF

= −4ΠW+W−(0) , (D9)

as in [62]. S and T lend their name from the well known Peskin-Takeuchi parameters S and

T which are related to the new ones via [55, 62]:

αS

4s2W= S − Y −W , αT = T − s2W

1− s2WY . (D10)

Here α is the electromagnetic structure constant and s2W is the weak mixing angle. Therefore

in the case where W = Y = 0 we have the simple relation

S =αS

4s2W, T = αT . (D11)

36

In our model these parameters read:

S =(2− χ)χg2

2g2 + (2− 2χ+ χ2)g2, (D12)

T = 0 , (D13)

W = M2W

g2(M2A + (χ− 1)2M2

V )

(2g2 + (2 + (χ− 2)χ)g2)M2AM

2V

, (D14)

Y = M2W

g′2(5M2A + (χ− 1)2M2

V )

(2g2 + (6 + (χ− 2)χ)g′2)M2AM

2V

, (D15)

U = 0 , (D16)

V = 0 , (D17)

X = g g′M2

W

M2AM

2V

M2A − (χ− 1)2M2

V√(2g2 + (2− 2χ+ χ2)g2)(2g2 + (6− 2χ+ χ2)g′2)

. (D18)

In these expressions the coupling constants g, g′ and g are the ones in the Lagrangian

associated to the yet to be diagonalized spin one states. W , Y and X are sensitive to the

ratio M2W/M

2 with M2 the lightest of the massive spin one fields. We have checked that

even taking an axial vector mass as small as 500 GeV while keeping large g for fixed S of

order 0.1 one is able to satisfy the experimental constraints on all of the parameters.

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[65] Small splittings with respect to the electroweak scale will be induced by the standard model

41

corrections per se.

[66] For the MWT we separate the contribution due to the new leptonic sector (which will be dealt

with later in the main text) and the one due to the underlying strongly coupled gauge theory

which is expected to be well represented by the perturbative contribution and is of the order

of 1/2π. When comparing with the S parameter from the vector meson sector of the effective

theory we should subtract from the underlying S the one due to the new fermionic composite

states U and D. This contribution is very small since it is 1/6π in the limit when these states

are degenerate and heavier than the zed gauge boson.

42